Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

Marcello Lucia; Guido Sweers

doi:10.3934/cpaa.2021152

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2021, Volume 20, Issue 12: 4177-4193. Doi: 10.3934/cpaa.2021152

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Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

Marcello Lucia^1, and
Guido Sweers^2, ,

1.
The City University of New York, CSI, Mathematics Department, Staten Island, New York 10314, USA
2.
University of Cologne, Dept. of Mathematics & Computer Science, 50931 Cologne, Germany

^* Corresponding author
^* Corresponding author

Received: April 30, 2021

Early access: September 2021

Published: December 2021

The first author was supported by the Alexander von Humboldt foundation and by MINECO grant MTM2017-84214-C2-1-P

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Abstract

We consider fully coupled cooperative systems on $ \mathbb{R}^n $ with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on $ \mathbb{R}^2 $ with exponential nonlinearity, are nondegenerate.

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Mathematics Subject Classification: Primary: 58J05, 35B40; Secondary: 58J28.

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Figure 1. Typical behaviour of the potential $ x\mapsto V_{i}\left( x\right) $ in $ L_{i} $

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References

[1]	R. A. Adams, Compact imbeddings of weighted Sobolev spaces on unbounded domains, J. Differ. Equ., 9 (1971), 325-334. doi: 10.1016/0022-0396(71)90085-4.
[2]	I. Birindelli, E. Mitidieri and G. Sweers, Existence of the principal eigenvalue for cooperative elliptic systems in a general domain, Differ. Equ., 35 (1999), 326-334.
[3]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[4]	H. Brezis and F. Merle, Uniform estimates and blow up behavior for solutions of $- \Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253. doi: 10.1080/03605309108820797.
[5]	K. J. Brown, C. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on$\mathbb{R}^{n}$, Proc. Amer. Math. Soc., 109 (1990), 147-155. doi: 10.2307/2048374.
[6]	J. Busca and B. Sirakov, Symmetry results for semi-linear elliptic systems in the whole space, J. Differ. Equ., 163 (2000), 41-56. doi: 10.1006/jdeq.1999.3701.
[7]	L. Cardoulis, Principal eigenvalues for systems of Schrödinger equations defined in the whole space with indefinite weights, Math. Slovaca, 65 (2015), 1079-1094. doi: 10.1515/ms-2015-0074.
[8]	K. C. Chang, Principal eigenvalue for weight matrix in elliptic systems, Nonlinear Anal., 46 (2001), 419-433. doi: 10.1016/S0362-546X(00)00140-1.
[9]	S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. Math., 99 (1974), 48-69. doi: 10.2307/1971013.
[10]	J. L. Chern, Z. Y. Chen and C. S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Commun. Math. Phys., 296 (2010), 323-351. doi: 10.1007/s00220-010-1021-z.
[11]	K. Choe, N. Kim, Y. Lee and C. S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in$\mathbb{R} ^{2}$, J. Funct. Anal., 273 (2017), 1734-1761. doi: 10.1016/j.jfa.2017.05.012.
[12]	J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integ. Equ. Operat. Theory, 2 (1979), 174-198. doi: 10.1007/BF01682733.
[13]	D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 49-52.
[14]	G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B, 345 (1995), 452-457. doi: 10.1016/0370-2693(94)01649-W.
[15]	X. Han and G. Huang, Existence theorems for a general $ 2\times 2$ non-Abelian Chern-Simons-Higgs system over a torus, J. Differ. Equ., 263 (2017), 1522-1551. doi: 10.1016/j.jde.2017.03.017.
[16]	P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions,, in Nonlinear Analysis and Optimization, Springer, Berlin, 1984. doi: 10.1007/BFb0101496.
[17]	R. Janssen, The dirichlet problem for second order elliptic operators on unbounded domains, Appl. Anal., 19 (1985), 201-216. doi: 10.1080/00036818508839545.
[18]	C. Kim, C. Lee, P. Ko, B. H. Lee and H. Min, Schrödinger fields on the plane with $[U(1)]^N$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840. doi: 10.1103/PhysRevD.48.1821.
[19]	C. S. Lin, A. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350. doi: 10.1016/j.jfa.2007.03.010.
[20]	C. S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347. doi: 10.1007/s00220-009-0774-8.
[21]	E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286. doi: 10.1002/mana.19951730115.
[22]	B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153. doi: 10.1007/BF01162028.
[23]	C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
[24]	R. G. Pinsky, Positive Harmonic Functions and Diffusion,, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511526244.
[25]	M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bull. Amer. Math. Soc., 72 (1966), 251-255. doi: 10.1090/S0002-9904-1966-11485-4.
[26]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, , Prentice-Hall, Inc., Englewood Cliffs, N. J. 1967
[27]	B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.
[28]	H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.
[29]	G. Sweers, Strong positivity in $C(\bar{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271. doi: 10.1007/BF02570833.
[30]	G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217. doi: 10.1007/s00526-006-0062-9.
[31]	Y. Yang, The relativistic non-abelian Chern-Simons equations, Commun. Math. Phys., 186 (1997), 199-218. doi: 10.1007/BF02885678.